This section provides a physical interpretation of equations 3.7, which will lead to a deeper understanding of the mechanics of inversion.
The causal Green's function gc and the
acausal Green's ga function for a point source in a 3-D
homogeneous medium are given by:
Note that these Green's functions are time invariant, which means that their values depend on the time lag between the source time ts and listening time t. This is useful, because it means that a CSG collected in May should look the same as one collected in July for the same recording geometry and experimental site. In general, the Green's function for acoustic fields are time-invariant in an arbitrary velocity model (Morse and Feshbach, 1953).
Figure 3.5 shows that these Green's functions describe either a backward or forward light cone, where the apex of the cone kisses the source point at the time ts. For a buried point source each cone intersects the surface plane z=0 along an hyperbola. The causal Green's function emulates exploding wavefronts from a point "source", while the acausal Green's function emulates imploding wavefronts from a point "sink".
Taking the Fourier transform of
equation 3.11 w/r to the